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Encyclopedia of Complexity and Systems Science pp 1–39Cite as

Cooperative Games (von Neumann–Morgenstern Stable Sets)

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Definition of the Subject

The von Neumann–Morgenstern stable set (hereafter stable set) is the first solution concept in cooperative game theory defined by J. von Neumann and O. Morgenstern. Though it was defined cooperative games in characteristic function form, von Neumann and Morgenstern gave a more general definition of a stable set in abstract games. Later, J. Greenberg and M. Chwe cleared a way to apply the stable set concept to the analysis of noncooperative games in strategic and extensive forms. Stable sets in a characteristic function form game may not exist, as was shown by W. F. Lucas for a ten-person game that does not admit a stable set. On the other hand, stable sets exist in many important games. In voting games, for example, stable sets exist, and they indicate what coalitions can be formed in detail. The core, on the other hand, can be empty in voting games, though it is one of the best known solution concepts in cooperative game theory. The analysis of stable sets is...

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Abbreviations

Abstract game:

An abstract game consists of a set of outcomes and a binary relation, called domination, on the outcomes. Von Neumann and Morgenstern presented this game form for general applications of stable sets.

Characteristic function form game:

A characteristic function form game consists of a set of players and a characteristic function that gives each group of players, called a coalition, a value or a set of payoff vectors that they can gain by themselves. It is a typical representation of cooperative games. For characteristic function form games, several solution concepts are defined such as von Neumann–Morgenstern stable set, core, bargaining set, kernel, nucleolus, and Shapley value.

Domination:

Domination is a binary relation defined on the set of imputations, outcomes, or strategy combinations, depending on the form of a given game. In characteristic function form games, an imputation is said to dominate another imputation if there is a coalition of players such that they can realize their payoffs in the former imputation by themselves and make each of them better off than in the latter. Domination given a priori in abstract games can be also interpreted in the same way. In strategic form games, domination is defined on the basis of commonly beneficial changes of strategies by coalitions.

External stability:

A set of imputations (outcomes, strategy combinations) satisfies external stability if any imputation outside the set is dominated by some imputation inside the set.

Farsighted stable set:

A farsighted stable set is a von Neumann–Morgenstern stable set defined by indirect domination. That is, a farsighted stable set satisfies internal stability and external stability with respect to indirect domination.

Imputation:

An imputation is a payoff vector in a characteristic function form game that satisfies group rationality and individual rationality. The former means that the players divide the amount that the grand coalition of all players can gain, and the latter says that each player is assigned at least the amount that he/she can gain by him/herself.

Indirect domination:

Indirect domination is a domination relation that takes into account the farsightedness of the players. Intuitively, an outcome is indirectly dominated by another if the latter can be reached through a sequence of deviations, and the coalitions when making their moves in the deviations are made better off at the final outcome than at the outcome when they deviate.

Internal stability:

A set of imputations (outcomes, strategy combinations) satisfies internal stability if there is no domination between any two imputations in the set.

Strategic form game:

A strategic form game consists of a player set, each player’s strategy set, and each player’s payoff function. It is usually used to represent noncooperative games.

Von Neumann– Morgenstern stable set:

A set of imputations (outcomes, strategy combinations) is a von Neumann–Morgenstern stable set if it satisfies both internal and external stability.

Bibliography

  • Aumann R, Peleg B (1960) Von Neumann-Morgenstern solutions to cooperative games without side payments. Bull Am Math Soc 66:173–179

    Article  MATH  MathSciNet  Google Scholar 

  • Bala V, Goyal S (2000) A noncooperative model of network formation. Econometric 638:1181–1229

    Article  MathSciNet  Google Scholar 

  • Baneijee S, Konishi H, Sonmez T (2001) Core in a simple coalition formation game. Soc Choice Welf 18:135–153

    Article  MathSciNet  Google Scholar 

  • Barberà S, Gerber A (2003) On coalition formation: durable coalition structures. Math Soc Sci 45:185–203

    Article  MATH  Google Scholar 

  • Beal S, Durieu J, Solal P (2008) Farsighted coalitional stability in TU-games. Math Soc Sci 56:303–313

    Article  MATH  MathSciNet  Google Scholar 

  • Bemheim D, Peleg B, Whinston M (1987) Coalition-proof Nash equilibria. J Econ Theory 42:1–12

    Article  Google Scholar 

  • Bhattacharya A, Brosi V (2011) An existence result for farsighted stable sets of games in characteristic function form. Int J Game Theory 40:393–401

    Article  MATH  MathSciNet  Google Scholar 

  • Bogomolnaia A, Jackson MO (2002) The stability of hedonic coalition structures. Games Econ Behav 38:201–230

    Article  MATH  MathSciNet  Google Scholar 

  • Bott R (1953) Symmetric solutions to majority games. In: Kuhn HW, Tucker AW (eds) Contribution to the theory of games, vol II. Annals of Mathematics Studies, vol 28. Princeton University Press, Princeton, pp 319–323

    Google Scholar 

  • Chwe MS-Y (1994) Farsighted coalitional stability. J Econ Theory 63:299–325

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Diamantoudi E (2005) Stable cartels revisited. Econ Theory 26:907–921

    Article  MATH  MathSciNet  Google Scholar 

  • Diamantoudi E, Xue L (2003) Farsighted stability in hedonic games. Soc Choice Welf 21:39–61

    Article  MATH  MathSciNet  Google Scholar 

  • Diamantoudi E, Xue L (2007) Coalitions, agreements and efficiency. J Econ Theory 136:105–125

    Article  MATH  MathSciNet  Google Scholar 

  • Ehlers L (2007) Von Neumann-Morgenstem stable sets in matching problems. J Econ Theory 134:537–547

    Article  MATH  MathSciNet  Google Scholar 

  • Einy E, Shitovitz B (1996) Convex games and stable sets. Games Econ Behav 16:192–201

    Article  MATH  MathSciNet  Google Scholar 

  • Einy E, Holzman R, Monderer D, Shitovitz B (1996) Core and stable sets of large games arising in economics. J Econ Theory 68:200–211

    Article  MATH  MathSciNet  Google Scholar 

  • Funaki Y, Yamato T (2014) Stable coalition structures under restricted coalitional changes. Int Game Theory Rev 16 doi: 10.1142/S0219198914500066

    Google Scholar 

  • Gale D, Shapley LS (1962) College admissions and the stability of marriage. Am Math Mon 69:9–15

    Article  MATH  MathSciNet  Google Scholar 

  • Greenberg J (1990) The theory of social situations: an alternative game theoretic approach. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Greenberg J, Monderer D, Shitovitz B (1996) Multistage situations. Econometrica 64:1415–1437

    Article  MATH  MathSciNet  Google Scholar 

  • Greenberg J, Luo X, Oladi R, Shitovitz B (2002) (Sophisticated) stable sets in exchange economies. Games Econ Behav 39:54–70

    Article  MATH  MathSciNet  Google Scholar 

  • Griesmer JH (1959) Extreme games with three values. In: Tucker AW, Luce RD (eds) Contribution to the theory of games, vol IV. Annals of Mathematics Studies, vol 40. Princeton University Press, Princeton, pp 189–212

    Google Scholar 

  • Gusfield D, Irving R (1989) The stable marriage problem: structure and algorithms. MIT Press, Boston

    MATH  Google Scholar 

  • Harsanyi J (1974) An equilibrium-point interpretation of stable sets and a proposed alternative definition. Manag Sci 20:1472–1495

    Article  MATH  MathSciNet  Google Scholar 

  • Hart S (1973) Symmetric solutions of some production economies. Int J Game Theory 2:53–62

    Article  MATH  Google Scholar 

  • Hart S (1974) Formation of cartels in large markets. J Econ Theory 7:453–466

    Article  Google Scholar 

  • Heijmans J (1991) Discriminatory von Neumann-Morgenstem solutions. Games Econ Behav 3:438–452

    Article  MATH  MathSciNet  Google Scholar 

  • Herings JJ, Mauleon A, Vannetelbosch V (2009) Farsightedly stable networks. Games Econ Behav 67:526–541

    Article  MATH  MathSciNet  Google Scholar 

  • Herings JJ, Mauleon A, Vannetelbosch V (2010) Coalition formation among farsighted agents. Games 1:286–298

    Article  MathSciNet  Google Scholar 

  • Jackson MO, van den Nouweland A (2005) Strongly stable networks. Games Econ Behav 51:420–444

    Article  MATH  Google Scholar 

  • Jackson MO, Wolinsky A (1996) A strategic model of social and economic networks. J Econ Theory 71:44–74

    Article  MATH  MathSciNet  Google Scholar 

  • Kamijo Y, Muto S (2010) Farsighted coalitional stability of a price leadership cartel. Jpn Econ Rev 61:455–465

    Article  MathSciNet  Google Scholar 

  • Kaneko M (1987) The conventionally stable sets in noncooperative games with limited observations I: definition and introductory argument. Math Soc Sci 13:93–128

    Article  MATH  Google Scholar 

  • Kawasaki R (2010) Farsighted stability of the competitive allocations in an exchange economy with indivisible goods. Math Soc Sci 59:46–52

    Article  MATH  MathSciNet  Google Scholar 

  • Kawasaki R, Muto S (2009) Farsighted stability in provision of perfectly “lumpy” public goods. Math Soc Sci 58:98–109

    Article  MATH  MathSciNet  Google Scholar 

  • Klaus B, Klijn F, Walzl M (2010) Farsighted house allocation. J Math Econ 46:817–824

    Article  MATH  MathSciNet  Google Scholar 

  • Klaus B, Klijn F, Walzl M (2011) Farsighted stability for roommate markets. J Pub Econ Theory 13:921–933

    Article  Google Scholar 

  • Lucas WF (1968) A game with no solution. Bull Am Math Soc 74:237–239

    Article  MATH  Google Scholar 

  • Lucas WF (1990) Developments in stable set theory. In: Ichiishi T et al (eds) Game theory and applications. Academic, New York, pp 300–316

    Google Scholar 

  • Lucas WF, Rabie M (1982) Games with no solutions and empty cores. Math Oper Res 7:491–500

    Article  MATH  MathSciNet  Google Scholar 

  • Lucas WF, Michaelis K, Muto S, Rabie M (1982) A new family of finite solutions. Int J Game Theory 11:117–127

    Article  MATH  MathSciNet  Google Scholar 

  • Luo X (2001) General systems and cp-stable sets: a formal analysis of socioeconomic environments. J Math Econ 36:95–109

    Article  MATH  Google Scholar 

  • Luo X (2009) On the foundation of stability. Econ Theory 40:185–201

    Article  ADS  MATH  Google Scholar 

  • Manlove D (2013) Algorithmics of matching under preferences. World Scientific, Singapore

    Book  MATH  Google Scholar 

  • Mariotti M (1997) A model of agreements in strategic form games. J Econ Theory 74:196–217

    Article  MATH  MathSciNet  Google Scholar 

  • Masuda T (2002) Farsighted stability in average return games. Math Soc Sci 44:169–181

    Article  MATH  MathSciNet  Google Scholar 

  • Mauleon A, Vannetelbosch V, Vergote W (2011) VonNeumann -Morgenstem farsightedly stable sets in two-sided matching. Theoretical Econ 6:499–521

    Article  MATH  MathSciNet  Google Scholar 

  • Moulin H (1995) Cooperative microeconomics: a game-theoretic introduction. Princeton University Press, Princeton

    Book  Google Scholar 

  • Muto S (1979) Symmetric solutions for symmetric constant-sum extreme games with four values. Int J Game Theory 8:115–123

    Article  MATH  MathSciNet  Google Scholar 

  • Muto S (1982a) On Hart production games. Math Oper Res 7:319–333

    Article  MATH  MathSciNet  Google Scholar 

  • Muto S (1982b) Symmetric solutions for (n, k) games. Int J Game Theory 11:195–201

    Article  MATH  MathSciNet  Google Scholar 

  • Muto S, Okada D (1996) Von Neumann-Morgenstem stable sets in a price-setting duopoly. Econ Econ 81:1–14

    Google Scholar 

  • Muto S, Okada D (1998) Von Neumann-Morgenstem stable sets in Cournot competition. Econ Econ 85:37–57

    Google Scholar 

  • Nakanishi N (1999) Reexamination of the international export quota game through the theory of social situations. Games Econ Behav 27:132–152

    Article  MATH  Google Scholar 

  • Nakanishi N (2001) On the existence and efficiency of the von Neumann-Morgenstem stable set in an n-player prisoner’s dilemma. Int J Game Theory 30:291–307

    Article  MATH  Google Scholar 

  • Nakanishi N (2009) Noncooperative farsighted stable set in an n-player prisoners’ dilemma. Int J Game Theory 38:249–261

    Article  MATH  Google Scholar 

  • Núñez M, Rafels C (2013) Von Neumann-Morgenstem solutions in the assignment market. J Econ Theory 148:1282–1291

    Article  MATH  Google Scholar 

  • Oladi R (2005) Stable tariffs and retaliation. Rev Int Econ 13:205–215

    Article  Google Scholar 

  • Owen G (1965) A class of discriminatory solutions to simple n-person games. Duke Math J 32:545–553

    Article  MATH  MathSciNet  Google Scholar 

  • Owen G (1968) n-Person games with only l, n-l, and n-person coalitions. Proc Am Math Soc 19:1258–1261

    MATH  Google Scholar 

  • Owen G (1995) Game theory, 3rd edn. Academic, New York

    MATH  Google Scholar 

  • Page FH, Wooders M (2009) Strategic basins of attraction, the path dominance core, and network formation games. Games Econ Behav 66:462–487

    Article  MATH  MathSciNet  Google Scholar 

  • Page FH, Wooders MH, Kamat S (2005) Networks and farsighted stability. J Econ Theory 120:257–269

    Article  MATH  MathSciNet  Google Scholar 

  • Peleg B (1986) A proof that the core of an ordinal convex game is a von Neumann-Morgenstem solution. Math Soc Sci 11:83–87

    Article  MATH  MathSciNet  Google Scholar 

  • Quint T, Wako J (2004) On house swapping, the strict core, segmentation, and linear programming. Math Oper Res 29:861–877

    Article  MATH  MathSciNet  Google Scholar 

  • Ray D, Vohra R (1997) Equilibrium binding agreements. J Econ Theory 73:30–78

    Article  MATH  MathSciNet  Google Scholar 

  • Rosenmüller J (1977) Extreme games and their solutions. Lecture notes in economics and mathematical systems, vol 145. Springer, Berlin

    Google Scholar 

  • Rosenmüller J, Shitovitz B (2000) A characterization of vNM-stable sets for linear production games. Int J Game Theory 29(3):9–61

    Google Scholar 

  • Roth AE, Postlewaite A (1977) Weak versus strong domination in a market with indivisible goods. J Math Econ 4:131–137

    Article  MATH  MathSciNet  Google Scholar 

  • Roth AE, Sotomayor MO (1990) Two-sided matching: a study in game-theoretic modeling and analysis. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Roth AE, Vande Vate JH (1990) Random paths to stability in two-sided matching. Econometrica 58:1475–1480

    Article  MATH  MathSciNet  Google Scholar 

  • Serrano R, Volij O (2008) Mistakes in cooperation: the stochastic stability of Edgeworth’s recontracting. Econ J 118:1719–1741

    Article  Google Scholar 

  • Shapley LS (1953) Quota solutions of n-person games. In: Kuhn HW, Tucker TW (eds) Contribution to the theory of games, vol II. Annals of Mathematics Studies, vol 28. Princeton University Press, Princeton, pp 343–359

    Google Scholar 

  • Shapley LS (1959) The solutions of a symmetric market game. In: Tucker AW, Luce RD (eds) Contribution to the theory of games, vol IV. Annals of Mathematics Studies, vol 40. Princeton University Press, Princeton, pp 145–162

    Google Scholar 

  • Shapley LS (1962) Simple games: an outline of the descriptive theory. Behav Sci 7:59–66

    Article  MathSciNet  Google Scholar 

  • Shapley LS (1964) Solutions of compound simple games. In: Tucker AW et al (eds) Advances in game theory. Annals of Mathematics Studies, vol 52. Princeton University Press, Princeton, pp 267–305

    Google Scholar 

  • Shapley LS (1971) Cores of convex games. Int J Game Theory 1:11–26

    Article  MATH  MathSciNet  Google Scholar 

  • Shapley LS, Scarf H (1974) On cores and indivisibilities. J Math Econ 1:23–37

    Article  MATH  MathSciNet  Google Scholar 

  • Shapley LS, Shubik M (1972) The assignment game I: the core. Int J Game Theory 1:111–130

    Article  MathSciNet  Google Scholar 

  • Shino J, Kawasaki R (2012) Farsighted stable sets in Hotelling’s location games. Math Soc Sci 63:23–30

    Article  MATH  MathSciNet  Google Scholar 

  • Shitovitz B, Weber S (1997) The graph of Lindahl correspondence as the unique von Neumann-Morgenstem abstract stable set. J Math Econ 27:375–387

    Article  MATH  MathSciNet  Google Scholar 

  • Shubik M (1985) A game-theoretic approach to political economy. MIT Press, Boston

    Google Scholar 

  • Simonnard M (1966) Linear programming. Prentice-Hall, New Jersey

    MATH  Google Scholar 

  • Solymosi T, Raghavan TES (2001) Assignment games with stable core. Int J Game Theory 30:177–185

    Article  MATH  MathSciNet  Google Scholar 

  • Sung SC, Dimitrov D (2007) On myopic stability concepts for hedonic games. Theory Dec 62:31–45

    Article  MATH  MathSciNet  Google Scholar 

  • Suzuki A, Muto S (2000) Farsighted stability in prisoner’s dilemma. J Oper Res Soc Jpn 43:249–265

    ADS  MATH  MathSciNet  Google Scholar 

  • Suzuki A, Muto S (2005) Farsighted stability in n-person prisoner’s dilemma. Int J Game Theory 33:431–445

    Article  MATH  MathSciNet  Google Scholar 

  • Suzuki A, Muto S (2006) Farsighted behavior leads to efficiency in duopoly markets. In: Haurie A et al (eds) Advances in dynamic games. Birkhauser, Boston, pp 379–395

    Chapter  Google Scholar 

  • Toda M (1997) Implementation and characterizations of the competitive solution with indivisibility. Mimeo

    Google Scholar 

  • von Neumann J, Morgenstem O (1953) Theory of games and economic behavior, 3rd edn. Princeton University Press, Princeton

    MATH  Google Scholar 

  • Wako J (1984) A note on the strong core of a market with indivisible goods. J Math Econ 13:189–194

    Article  MATH  MathSciNet  Google Scholar 

  • Wako J (1991) Some properties of weak domination in an exchange market with indivisible goods. Jpn Econ Rev 42:303–314

    Google Scholar 

  • Wako J (1999) Coalitional-proofness of the competitive allocations in an indivisible goods market. Fields Inst Commun 23:277–283

    MathSciNet  Google Scholar 

  • Wako J (2010) A polynomial-time algorithm to find von Neumann-Morgenstem stable matchings in marriage games. Algorithmica 58:188–220

    Article  MATH  MathSciNet  Google Scholar 

  • Xue L (1997) Nonemptiness of the largest consistent set. J Econ Theory 73:453–459

    Article  MATH  Google Scholar 

  • Xue L (1998) Coalitional stability under perfect foresight. Econ Theory 11:603–627

    Article  MATH  Google Scholar 

Books and Reviews

  • Lucas WF (1992) Von Neumann-Morgenstem stable sets. In: Aumann RJ, Hart S (eds) Handbook of game theory with economic applications, vol 1. North-Holland, pp 543–590

    Google Scholar 

  • Shubik M (1982) Game theory in the social sciences: concepts and solutions. MIT Press, Boston

    MATH  Google Scholar 

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Authors and Affiliations

  1. Graduate School of Decision Science and Technology, Tokyo Institute of Technology, Tokyo, Japan

    Ryo Kawasaki & Shigeo Muto

  2. Department of Economics, Gakushuin University, Tokyo, Japan

    Jun Wako

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  1. Ryo Kawasaki
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Correspondence to Ryo Kawasaki .

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    Robert A. Meyers

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Kawasaki, R., Wako, J., Muto, S. (2015). Cooperative Games (von Neumann–Morgenstern Stable Sets). In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_99-2

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  1. Latest

    Cooperative Games (Von Neumann-Morgenstern Stable Sets)
    Published:
    10 April 2018

    DOI: https://doi.org/10.1007/978-3-642-27737-5_99-3

  2. Original

    Cooperative Games (von Neumann–Morgenstern Stable Sets)
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    30 November 2015

    DOI: https://doi.org/10.1007/978-3-642-27737-5_99-2

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